We report on the detection of Ne VIII in the Hubble Space Telescope/Cosmic Origins Spectrograph (COS) spectrum of the intervening absorption system at z = 0.495096 toward PKS 0405-123 (z em = 0.5726). The high signal-to-noise COS spectrum also covers absorption from H I, C III, O III, O IV, and O VI associated with this multiphase system. The Ne VIII is detected with high significance in both lines of the doublet, with integrated column densities of log Na (Ne VIII 770) = 13.96 ± 0.06 and log Na (Ne VIII 780) = 14.08 ± 0.07. We find the origin of Ne VIII consistent with collisionally ionized gas at T ~ 5 × 105 K with a large baryonic column density of N(H) ~ 1019-1020 cm-2. The metallicity in the Ne VIII gas phase is estimated to be [Ne/H] ~-0.6 ± 0.3 dex. The intermediate ions such as C III, O III, O IV, and H I are consistent with photoionization in lower ionization gas at T ~ 104 K. The O V and O VI in this absorber can have contributions from both the photoionized and collisionally ionized gas phases. The absorber is at |Δv| = 180 km s-1 systematic velocity and ρ = 110h -1 70 kpc projected separation from an MR = -19.6 galaxy of extended morphology. The collisionally ionized gas at T ~ 5 × 105 K detected in Ne VIII and O VI points to an origin in multiphase gas embedded in the hot halo of the galaxy, or in a nearby WHIM structure. The high-sensitivity UV spectroscopy afforded by COS has opened up new opportunities for discovering large reservoirs of missing baryons in the low-z universe through the detection of Ne VIII systems. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 05-26555, and the NASA-CNES/ESA Far-Ultraviolet Spectroscopic Explorer mission, operated by the Johns Hopkins University, supported by NASA contract NAS 05-32985.